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A Threshold Equation for Action Potential Initiation

Figure 2

Relationships between spike threshold definitions.

A, Excitability curve of the neuron model (dV/dt = (F(V)+I)/C; see Materials and Methods) for DC input current I = 0 (solid curve) and (dashed curve). With I = 0, the lower equilibrium (filled circle) corresponds to the resting potential Vr, while the higher equilibrium (open circle) corresponds to the spike threshold with short pulses (as in Fig. 1A): if the membrane potential is quickly shifted above , the membrane potential blows up and the neuron spikes (thus, this corresponds to the case when , i.e., an impulse current). Slowly increasing the input current amounts to vertically shifting the excitability curve, and the membrane potential follows the resting equilibrium until it disappears, when . The voltage VT at that point corresponds to the minimum of the excitability curve. The empirical threshold (with the first derivative method) is the voltage at the intersection of the excitability curve with the horizontal line dV/dt = kth (dashed line). The slope threshold corresponds to the radius of curvature at VT. B, Threshold for short pulses (solid line) and empirical threshold (blue dashed line) as a function of the threshold for slow inputs VT (black dashed line is the identity line): the definitions are quantitatively different but highly correlated. C, Dependence of empirical threshold on derivative criterion kth: spike onsets are measured on a voltage trace (as in Fig. 1C) with derivative criterion kth = 7.5 mV/ms (blue dots), 10 mV/ms (black), 12.5 mV/ms (green) and 15 mV/ms (red). D, Empirical threshold measured with kth = 7.5 mV/ms (blue dots), 12.5 mV/ms (green) and 15 mV/ms (red) vs. threshold measured with 10 mV/ms, and linear regression lines. The dashed line represents the identity. The value of the derivative criterion (kth) impacts the threshold measure but not its relative variations.

Figure 2